On the cyclotomic Iwasawa invariants of elliptic curves of rank one

TL;DR

This paper develops an explicit numerical criterion to determine when the Iwasawa invariants of a rank one elliptic curve over Q reach their minimal possible sum, extending previous criteria to both ordinary and supersingular primes.

Contribution

It introduces a new explicit criterion, comparable to Gold's, for assessing the minimality of Iwasawa invariants of elliptic curves at good primes.

Findings

Criterion applicable to both ordinary and supersingular primes.

Provides a practical method to compute Iwasawa invariants.

Enhances understanding of elliptic curve invariants in Iwasawa theory.

Abstract

Fix an elliptic curve $E$ over $\mathbb Q$ of rank $1$. In this paper, we develop an explicit numerical criterion, comparable to Gold's criterion, that determines whether the Iwasawa invariants of the elliptic curve at a good (ordinary or supersingular) prime attain their smallest possible value, i.e. whether $\mu_p^\pm(E) +\lambda^\pm_p(E)=1$ in the supersingular case or $\mu_p(E) + \lambda_p(E)=1$ in the ordinary case.